Why are a few chess openings named King's Indian Defense, Queen's Indian Defense, Nimzo-Indian defense, and so on?
What is the significance of Indian in all these opening names? What's India's contribution to it?
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In Indian Chess, the game that was played in the 18th, 19th and early 20th centuries in India (not to be confused with its ancestor Chaturanga), the rules allowed no castle, but double-step pawn moves were not allowed and:
The king can make a knight's move once in a game, known as Indian castling.
As a consequence, g3 followed by Kg2 for White or ...b6 followed by ...Kb7 (the black king stands on d8 at the beginning of the game in Indian chess) were very common moves in the opening, since it was the fastest way to bring the king towards the corner where it will usually find the most safety.
Of course, playing g3, b3 or ...b6, ...g6 could also be followed by a "normal" fianchetto with the camel (our bishop).
When Indian players started playing western chess (during colonization), they quite naturally used a lot of fianchettoes, since they are common in Indian Chess. As explained in user1583209's answer, that was the case for Moheschunder Bannerjee in his games against Cochrane around 1850:
The Indian Defences by P-KKt3 coupled with P-Q3, or P-QKt3 coupled with P-K3, were largely taught to European players by the example of Moheschunder and other Indians, to whom the fianchetto developments were a natural legacy from their own game. It is notable that many of Bannerjee's openings were based on single pawn moves, the "legacy" mentioned by Sergeant, as in the rules of Indian chess.
The Chess Player's Chronicle used the term "Indian defense" to refer to 1.e4 d6 2.d4 g6 after him about 30 years later (1884). This is the first documented use of the appelation about that kind of defense with fianchetto - although "Indian Opening" appeared earlier to refer to 1.e4 e5 2.d3.
The term "Indian Defense" as we understand it today (After 1.d4 Nf6, thus excluding Pirc-Ufimsev and other schemes with e4 and not c4 for White) established itself after WWI, its use being popularized by Savielly Tartakower and Hans Kmoch.
Chess has evolved from the Indian game Shatranj. In early versions of Shatranj, the starting positions included the fianchetto, which is common in all of the modern chess openings you listed. This is why these openings have the word Indian in them.
In "Indian" defenses, Black holds back his center pawns, especially his d-pawn, against a White 1. d4. This "Indian" style is in contrast to the European style of opposing 1. d4 with 1. ... d5, and 1. e4 with 1. ... e5.
Instead, Black opts to contest White's control of the center from the side. In the King's and Queen's Indian defenses, this means moving g6 or b6 respectively, and "fianchettoing" a bishop along the "long" diagonal. In the Nimzo-Indian defense (Nimzo stands for Aron Nimzowich), it means playing B b4 after e6. But the idea is to oppose White's d4 with c5 and not d5.
Indian Castling is derived from Shatranjan ( Not Shatranj ). ShadYantra is the actual game from which more advanced form of game called Shatranjan was derived by Kautilya (*Chanakya) some 2300 yrs ago. Chaturanga and Shatranj are variants.
In Shatranjan, Kings can move like Knights with the help of Royal and Semi Royal Aids. But in modern chess there are no Royal and Semi Royal pieces.
Apart from the name Indian we have French defence, Yugoslav attack of Sicilian, the Russian attack of some defence which I do not remember right now but, the names came from the country where they originated but they are played in their respective way.
Chess has its origination in India and it was named Chaturanga.
In the medieval world, India was the only country who deployed Elephants during warfare so we have Rooks in the Game. The Indian Name in many Openings does not necessarily mean a Fianchetto of the Bishop as we have Nimzo-Indian Defence where Bishop comes to b4 but it's the introduction of a Bishop which is also an Indian animal Camel which was used in north India during Warfare.
So I believe that would be the only idea behind this naming convention. Even I have tried finding the reasoning but there is nothing specific. It is a sad history of chess that we do not have the name of the Person who invented the Game.